2011
DOI: 10.1007/s00211-011-0407-y
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Interlacing of zeros of Gegenbauer polynomials of non-consecutive degree from different sequences

Abstract: A theorem due to Stieltjes' states that if { p n } ∞ n=0 is any orthogonal sequence then, between any two consecutive zeros of p k , there is at least one zero of p n whenever k < n, a property called Stieltjes interlacing. We show that Stieltjes interlacing extends to the zeros of Gegenbauer polynomials C λ n+1 and C λ+t n−1 , λ > − 1 2 , if 0 < t ≤ k + 1, and also to the zeros of C λ n+1 and C λ+k n−2 if k ∈ {1, 2, 3}. More generally, we prove that Stieltjes interlacing holds between the zeros of the kth der… Show more

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Cited by 8 publications
(6 citation statements)
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“…is Pochhammer's symbol. The smallest zero of the quadratic polynomial coefficient of L α n in (7) is (α + 2) 2 (3n + 2α + 5) − (α + 2) 2 (9(α + 2) 2 + 2(2α + 5)(α 2 + 5α + 10)n + (5α 2 + 25α + 38)n 2 ) 2(n + α + 2) 2 (8) and this is a sharper upper bound for the smallest zero of L α n+1 than the upper bound w n+1 < (α + 1)(α + 2)(α + 4)(2n + α + 3) (α + 1) 2 (α + 2) + (5α + 11)(n + 1)(n + α + 2)…”
Section: Applications To Classical Families and Comparison With Existmentioning
confidence: 99%
“…is Pochhammer's symbol. The smallest zero of the quadratic polynomial coefficient of L α n in (7) is (α + 2) 2 (3n + 2α + 5) − (α + 2) 2 (9(α + 2) 2 + 2(2α + 5)(α 2 + 5α + 10)n + (5α 2 + 25α + 38)n 2 ) 2(n + α + 2) 2 (8) and this is a sharper upper bound for the smallest zero of L α n+1 than the upper bound w n+1 < (α + 1)(α + 2)(α + 4)(2n + α + 3) (α + 1) 2 (α + 2) + (5α + 11)(n + 1)(n + α + 2)…”
Section: Applications To Classical Families and Comparison With Existmentioning
confidence: 99%
“…We use the contiguous relation (12) and the 2 F 1 representation of the Kravchuk polynomials (6) to obtain the mixed three-term recurrence relation (n − 1)(N − n + 2) 2 (1 − p) 2 K n−2 (x; p, N + 1) = p(x + (n − 1)p − N − 1)K n (x; p, N ) − P 2 (x)K n−1 (x; p, N )…”
Section: Proofsmentioning
confidence: 99%
“…We will refer to this as completed Stieltjes interlacing. An extension of this completed Stieltjes interlacing between polynomials belonging to the same orthogonal sequence, to polynomials from different orthogonal sequences, obtained by integer shifts of the appropriate parameter(s), was done in [6,7] and [8] for the Gegenbauer, Laguerre and Jacobi polynomials respectively.…”
Section: Introductionmentioning
confidence: 99%
“…Interlacing results for the zeros of different sequences of q-orthogonal sequences with shifted parameters are given in [19,26,35,40]. Completed Stieltjes interlacing of zeros of different orthogonal sequences was done for the Gegenbauer [14], Laguerre [16] and Jacobi polynomials [15] and apart from the papers cited in the previous paragraph, inner bounds for the extreme zeros of Gegenbauer, Laguerre and Jacobi polynomials were also given in [2,6,21,32,36,39]; bounds for the extreme zeros of the discrete orthogonal Charlier, Meixner, Krawtchouk and Hahn polynomials in [3,33], for the extreme zeros of the q-Jacobi and q-Laguerre polynomials in [21] and for the little q-Jacobi polynomials in [19]. Lower bounds for x n,1 and upper bounds for x n,n can be found in the case of classical continuous and discrete orthogonal polynomials in [1,3,11,12,22,30,32,33,39] and in [31], non-asymptotic bounds on the extreme zeros of (symmetric) orthogonal polynomials are given in terms of the coefficients of their three-term recurrence equations.…”
Section: Introductionmentioning
confidence: 99%